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Math Help - Find the probability density function

  1. #1
    Senior Member sfspitfire23's Avatar
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    Find the probability density function

    A point Q is chosen in the unit square. What is the density function of the sum of the coordinates? Of the product of the coordinates?

    Work-
    Let X be a r.v. on (0,1) and let Y be a r.v. on (0,1). These are independent.

    Then, I let V = X + Y.

    So, I then perform P(V < c) where c is just some constant (cumulative distribution function). I transform the inequality like so: P(Y < c - X).

    I feel like I'm very close to solving this but here is where I get stuck. What do I do now?
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Find the probability density function

    Quote Originally Posted by sfspitfire23 View Post
    A point Q is chosen in the unit square. What is the density function of the sum of the coordinates? Of the product of the coordinates?

    Work-
    Let X be a r.v. on (0,1) and let Y be a r.v. on (0,1). These are independent.

    Then, I let V = X + Y.

    So, I then perform P(V < c) where c is just some constant (cumulative distribution function). I transform the inequality like so: P(Y < c - X).

    I feel like I'm very close to solving this but here is where I get stuck. What do I do now?
    Hi sfspitfire23!

    The easiest way to do it, is to draw a unit square with a diagonal line representing the same values of v.
    The chance P(V < c) is then the area from the origin up to this line.
    This is a triangle for the first half and a diamond shaped area for the second half.
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  3. #3
    MHF Contributor
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    Re: Find the probability density function

    In general, if X is a nonnegative r.v. with pdf f(x) and Y is a nonnegative r.v. with pdf g(x), then P(X + Y\le c)=\int_0^c\int_0^{c-x}f(x)g(y)\,dy\,dx. In this case, the set of points (x, y) in the unit square such that x + y <= c is a filled right triangle with vertices (0, 0), (0, c) and (c, 0), and P(X + Y <= c) is the area of that triangle. Similarly, P(XY <= c) is the area below the hyperbola y = c / x.
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